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Dark Energy: Dark Energy:

Extending EinsteinExtending Einstein

Eric Linder University of California, BerkeleyLawrence Berkeley National Lab

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From Data to Theory (and back)From Data to Theory (and back)

To compare observations and theory we need a statistical measure of goodness of fit.

We need to compare the theory value, e.g. for distance-redshift,

dlum = (1+z) 0z dz’ / H(z’; m,w(z’) )

to the data Dlumi. For example 2 or likelihood

2 = i,j[Dlumi- dlum(zi)] COV-1(i,j) [Dlum

j- dlum(zj)]t

L = exp(- 2/2) [Gaussian near max likelihood]

We need 1) theory or robust parametrization w(z), 2) efficient method for estimating parameter errors given data characteristics.

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Fisher MatrixFisher Matrix

Fisher matrix gives lower limit for Gaussian likelihoods, quick and easy.

Fij = d2 (- ln L) / dpi dpj = O(dO/dpi) COV-1 (dO/dpj)

(pi) 1/(Fii)1/2

Example: O=dlum(z=0.1,0.2,…1), p=(m,w), COV=(d/d)d ij

Fw=k(dOk/d)(dOk/dw)k-2

2() COV(,w)

COV(,w) 2(w)C = F-1 =( )F Fw

Fw Fww

F = ( )Also called information matrix. Add independent data sets, or priors, by adding matrices.

e.g. Gaussian prior on m=0.280.03 via 2 = (m-0.28)2/0.032

See: Tegmark et al. astro-ph/9805117 Dodelson, “Modern Cosmology”

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Survival of the FittestSurvival of the Fittest

Fisher estimates give a N-dimension ellipsoid. Marginalize (integrate over the probability distribution) over parameters not of immediate interest by crossing out their row/column in F-1. Fix a parameter by crossing out row/column in F.

1 (68.3% probability enclosed) joint contours have d2=2.30 in 2-D (not d2=1). Read off 1 errors by projecting to axis and dividing by 1.52=2.30.

Orientation of ellipse shows degree of covariance (degeneracy).

Different types of observations can have different degeneracies (complementarity) and combine to give tight constraints.

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Model IndependenceModel Independence

We could check each theoretical model one by one against the data -- but there are 10x of them, each with their own parameters. We’d also like to predict / design results of different experiments.

Want model independent approach. Remember

H(z)=[m(1+z)3 + w exp{30z d ln(1+z) [1+w(z)]} ]1/2

Parametrize w(z). Keep close to the physics: both energy density and pressure enter the dynamics; directly related to kinetic/potential energy of scalar field.

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Model IndependenceModel Independence

Simplest parametrization, with physical dynamics,

w(a)=w0+wa(1-a)

Recall a=(1+z)-1.

Virtues:

• Model independent

• Excellent approximation to exact field equation solutions

• Robust against bias

• Well behaved at high z

Problems: Cannot handle rapid transitions or oscillations.

N.B.: constant w lacks important physics; w(z)=w0+w1z is Taylor expansion about low z only - pathological at high z.

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EigenmodesEigenmodes

w0, wa makes for easiest, robust comparison. But sometimes want nonparametric form.

Eigenmodes of w(z) give independent principal components (but depend on model, experiment, and probe).

Start with parameters of wi in z bins. Diagonalize Fisher matrix F=ETDE: D is diagonal, rows of E give eigenvectors.

w(z) = bi ei(z)

Localized eigenmodes L=ETD1/2E

Huterer & Cooray 2005

Huterer & Starkman 2003

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Design an ExperimentDesign an Experiment

Precision in measurement is not enough - one must beware degeneracies and systematics.

p2

p1

*

.

Degeneracy: e.g. Aw0+Bwa=const

Degeneracy: hypersurface, e.g. covariance with m

Systematic: offset error in data or model, e.g. evolution

or Systematic: floor to precision, e.g. calibration

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Mapping HistoryMapping History

Data over a range of redshifts can be effective at breaking degeneracies. Plus one gets leverage from a long baseline in expansion history.

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Controlling SystematicsControlling Systematics

Controlling systematics is the name of the game. Finding more objects is not.

Must understand the sources, instruments, and the theory interpretation.

Forthcoming experiments may deliver 100,000s of objects. But uncertainties do not reduce by 1/N.

Must choose cleanest probe, mature method, with multiple crosschecks.

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ComplementarityComplementarity

Complementarity of techniques (e.g. SN,WL,CMB,…)

• improves precision

• breaks degeneracies

• immunizes against systematics

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Design an ExperimentDesign an Experiment

How to design an experiment to explore dark energy?

•Choose clear, robust, mature techniques

•Rotate the contours thru choice of redshift span

•Narrow the contours thru systematics control

•Break degeneracies thru multiple probes

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Optimize an ExperimentOptimize an Experiment

Optimization depends on the question asked.

Recall that physics divided into 2 classes: thawing and freezing.

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Design an ExperimentDesign an Experiment

How to design an experiment to explore dark energy?

•Choose clear, robust, mature techniques

•Rotate the contours thru choice of redshift span

•Narrow the contours thru systematics control

•Break degeneracies thru multiple probes

With a strong experiment, we can even test the framework of physics.

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Acceleration = CurvatureAcceleration = Curvature

The Principle of Equivalence teaches that

Acceleration = Gravity = Curvature

Acceleration over time will get v=gh/c, so z = v/c = gh/c2 (gravitational redshift).

But, tt0 parallel lines not parallel (curvature)!

t0

t´Height

Time

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Dark energy is a completely unknown animal.

A new theory or a new component?

Track record:

Inner solar system motions General Relativity

Outer solar system motions Neptune

Galaxy rotation curves Dark Matter

Finding Our Way in the DarkFinding Our Way in the Dark

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Expansion HistoryExpansion History

Suppose we admit our ignorance:

H2 = (8/3) m + H2(a)

Effective equation of state:

w(a) = -1 - (1/3) dln (H2) / dln a

Modifications of the expansion history are equivalent to time variation w(a). Period.

Observations that map out expansion history a(t), or w(a), tell us about the fundamental physics of dark energy.

Alterations to Friedmann framework w(a)

gravitational extensions or high energy physics

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Expansion HistoryExpansion History

For modifications H2, define an effective scalar field with

V = (3MP2/8) H2 + (MP

2H02/16) [ d H2/d ln a]

K = - (MP2H0

2/16) [ d H2/d ln a]

Example: H2 = A(m)n

w = -1+n

Example: H2 = (8/3) [g(m) - m]

w= -1 + (g-1)/[ g/m - 1 ]

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The world is w(z)The world is w(z)

Don’t care if it’s braneworld, cardassian, vacuum metamorphosis, chaplygin, etc.

Simple, robust parametrization

w(a)=w0+wa(1-a)

Braneworld [DDG] vs. (w0,wa)=(-0.78,0.32)

Vacuum metamorph vs. (w0,wa)=(-1,-3)

Also agree on m(z) to 0.01 mag out to z=2

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Hidden DimensionsHidden Dimensions

Extra dimensions have been used for unification in physics since the 1920s.

Large extra dimensions -- braneworlds -- can be tested astronomically.

Spacetime is warped by e-y as one moves a distance y off a brane. Think of the spacetime properties as an index of refraction: such a spatial gradient n localizes light (and the rest of physics).

Gravity?Electro-magnetism?

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Warped GravityWarped Gravity

On large (cosmological distances) there may be leaking gravity. The cosmic expansion would appear slower over these distances, i.e. accelerating today!

Like localized light in a fiber optic, gravity will eventually leak off into hidden dimensions.

Or think of a tuning fork: it radiates sound in all directions, but the waves are stronger if localized.

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DGP Braneworld DynamicsDGP Braneworld Dynamics

More than 3 from flatness

rc = MPl2/(2M5

3)

rc = (2H0rc)-2

SNAP could determine rc to (rc)=0.003

Fairbairn & Goobar astro-ph/0511029

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Gravity Beyond 4DGravity Beyond 4D

z=1

z=2

z=3

=1/2

=1 (BW)

Can reproduce expansion or growth with quintessence, but not both.

DGP Braneworld, and H mods, obey freezer dynamics in w-w

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Revealing PhysicsRevealing Physics

• Time variation w(z) is a critical clue to fundamental physics. • Modifications of the expansion history = w(z).• But need an underlying theory - ? beyond Einstein gravity?

Growth history and expansion history work together.

Linder 2004, Phys. Rev. D 70, 023511 cf. Lue, Scoccimarro, Starkman Phys. Rev. D 69 (2004) 124015 for braneworld perturbations

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Growth HistoryGrowth History

Growth rate of density fluctuations g(a) = (m/m)/a

€

g + [5 + 12

d ln H 2

d ln a ] ′ g a−1 + [3+ 12

d ln H 2

d ln a − 32 G Ωm (a)] ga−2 = S(a)

€

g + [3 + 2ℜ] ′ g a−1 + [1+ 2ℜ − 32 G Ωm (a)] ga−2 = S(a)

€

g + [ 72 − 3

2 w(a)Ωφ (a)] ′ g a−1 + 32 [1− w(a)]GΩφ (a) ga−2 = S(a)

€

g + [4 − q] ′ g a−1 + [2 − q − 32 G Ωm (a)] ga−2 = S(a)

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Physics of GrowthPhysics of Growth

Growth g(a)=(/)/a depends purely on the expansion history H(z) -- and gravity theory.

€

g + [5 + 12

d ln H 2

d ln a ] ′ g a−1 + [3 + 12

d ln H 2

d ln a − 32 G Ωm (a)] ga−2 = S(a)

Expansion effects via w(z), but separate effects of gravity on growth.

g(a) = exp { 0ad ln a [m(a) -1] }

Growth index = 0.55+0.05[1+w(z=1)] Works to 0.05 – 0.2%!

0

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Growth and ExpansionGrowth and Expansion

Keep expansion history as w(z), growth deviation from expansion by .

With as free fit parameter, we can test framework, and the origin of dark energy.

Paradigm: To reveal the origin of dark energy, measure w, w, and . e.g. use SN+WL.

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Going NonlinearGoing Nonlinear

Efficient generation of grid of dark energy cosmologies.

Linder & White 2005 PRD 72, 061304(R)

Previous fit functions were only good to ~10% -- for . New technique is good to 1.5%, for general dark energy.

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Gravity’s BiasGravity’s Bias

Neglecting modified gravity will bias the cosmology unless gravity is properly accounted for (e.g. ).

Huterer & Linder 2006

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Going Beyond EinsteinGoing Beyond Einstein

To test Einstein gravity, we need growth and expansion. To test dark energy and GR, we need superb data.

9 parameter cosmology fit.

Testing GR via growth index degrades w0, wa by 15-25%.

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Fitting Beyond EinsteinFitting Beyond Einstein

How well can we fit gravity?

N.B. it’s important to include other effects on large scale structure such as m.

WL+SN+CMB can determine to 8%.

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Dark Energy SurprisesDark Energy Surprises

There is still much theoretical work needed!

Dark energy is…• Dark• Smooth on cluster scales• Accelerating

Maybe not completely! Clumpy in horizon? Maybe not forever!

It’s not quite so simple!

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Heart of DarknessHeart of Darkness

Is dark energy dark – only interacts gravitationally?

Self interaction: pseudoscalar quintessence

Coupling to matter: Chaplygin gas Leads to 5th force: limited by lab tests Unify dark energy with dark matter? Distorts matter power spectrum: ruled out unless within 10-5 of

Coupling to gravitation: Scalar-tensor theories = Extended quintessenceCan clump on subhorizon scales Can “turn on” from nonlinear structure formation?!

Higher dimension gravity: Scalaron quintessenceCan be written in terms of scalar-tensor and weff

Sandvik et al. 2003

The horror!

The horror!

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Theory and DataTheory and Data

Pinpointing Physics Is it ?dynamics via w

Checking Geometry allowing curvature

Testing GR new gravity

Thanks to Gary Bernstein, Dragan Huterer, Masahiro Takada for key contributions

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ComplementarityComplementarity

Cosmic acceleration is so revolutionary we need the crosschecks, synergy, reduced influence of systematics, robust answers of complementary probes.

SNAP space mission gives infrared and high redshift measurements, high resolution and lower systematics.

SNAP wide field telescope gives multiple probes (e.g. SN Ia, Weak Lensing, Clusters, Strong Lensing, SN II) and rich astronomical resources.

When you have a mystery ailment, you want a diagnosis with blood tests, EKG, MRI,...

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Frontiers of the UniverseFrontiers of the Universe

Breakthrough of the Year

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Cosmology holds the key to new physics in the next decade.

1998

2003 Let’s find out!